October 2006

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Math... What Good Is It?

by Donald Christiansen

This seems to be the question that many elementary school kids are asking. A teaching system that cannot answer them is often blamed for losing future engineers and scientists who, at this youthful stage, may be mathematically adept but uninterested in pursuing math studies because they cannot imagine any useful way to apply what they might learn.

It may be hard for engineers to understand why classroom math is disdained by young students. Math and mathematically soluble problems are intrinsically fascinating to us. And I think we sensed at an early age that mathematical tools are a necessity in solving complex technical problems in the real world.
My recollection is that in the primary grades there were lots of apples moving about among the Dick and Jane set, and that the usual problem was something along the lines of “How many apples does Jane have now?” Dick and Jane’s challenges were real-world enough for me, and I was happy to solve them.

Nevertheless, some evidence exists to support the view that with new teaching initiatives marginally interested students could be induced to develop a serious interest in formal mathematics at an early age.

Keith Devlin, the executive director of Stanford University’s Media X network, theorizes that students possess mathematical and computational instincts and that the only thing standing in the way of their improving their math skills is motivation and, of course, practice. The problem many people have with “school” arithmetic is that they never get to the meaning stage, says Devlin. It remains forever an abstract game of formal symbols.

He cites experiments in which young (subteen) street vendors in Brazil could, for example, quickly calculate the price of several coconuts based on their familiarity with the price of one. They used a common-sense, if sometimes convoluted process to do this. But given the same problem as a classroom exercise, devoid of context, they could not solve it, or would come up with an answer far off the mark. Not surprisingly, they did not care for formal math. The consequences of failing a classroom exercise were not equivalent to losing money in a street transaction. The stakes in the classroom were low.

An important step, Devlin thinks, is convincing students that abstract math is merely a formalized version of their innate mathematical abilities. I am not sure that all students, even gifted ones, can easily make the transition that this postulate implies. If you have had the opportunity to tutor young students in math, it is easy to conclude that some have a strong mathematical aptitude, and some don’t, motivation notwithstanding. And a few otherwise intellectually gifted adults have told me they consider themselves mathematically challenged.

Numbers don’t always count

Both kids and adults have a fascination with numbers — e.g., school grades, ball scores, lotteries, or stock market indexes. In Barnes & Noble, one can find “10001 Hints and Tips for the Home,” and 10 or 100 ways to do many other things. Yet this universal interest in the output of metrics and statistics has little or nothing to do with mathematics. There is a disconnect between numbers and abstract math, as apparently there is between students’ views of math and the real world.

We are also told that today’s students, and even adults, have difficulty remembering that 7 x 8 is 56, and not some other number, like 54, 45 or 64, and that it may take them some time to realize that 2 x 3 is not 5, confusing it with 2 + 3. Some linguists blame this on a phenomenon called pattern interference, but my guess is that the availability of calculators makes it unnecessary for them to be concerned with multiplication tables or adding in your head, as earlier generations did.

Moving on

At some point, the mathematically adept student embraces abstract manipulations and can take joy in math for math’s sake. In high school, our math teacher would add an optional, very difficult proof to our routine homework assignment. A few of us, most of whom later became engineers, would stay up until one o’clock in the morning if necessary (admitting this only to one another, not to the teacher) to solve it. We were not concerned that we could not relate the exercise to any real-world problem. I prefer to think that our classmates did not view us as showoffs, but rather as budding geniuses who would go on to create exciting new products through the magic of science and technology. We found solving these toughies fun, challenging and a point of honor. Sadly, we are told that today, in some schools, particularly in economically and socially disadvantaged areas, this would not be the case. Kids excelling in math or science would be ostracized and demeaned as out of the cultural mainstream.

Reconnecting

It is possible, as some educators propose, that the overwhelming exposure of youngsters to real-world problems and issues through television and the Internet preempts kids' interest in abstract studies. As one teenager put it, “Going on line totally impacts me!” One of the solutions, therefore, is to use these same media to link math and the real world.

Today, students and teachers alike can find a plethora of mathematical games and puzzles online. The many Web sites offer a wide range of challenges — for elementary grade schoolers and even post docs. The American Society for Engineering’s monthly e-newsletter, GO ENGINEERING, helps K-12 math teachers inject engineering-related material into their courses, to “make mathematics come alive.” One K-12 teacher resource is the TeachEngineering digital library, a partnership of Worcester Polytechnic Institute, Colorado School of Mines, University of Colorado at Boulder, Oregon State University, and ASEE.

With all the activity geared toward improving the teaching of math at the K-12 level, it is disappointing to learn that the outcomes are not improving, and in some cases deteriorating. Perhaps we are on the cusp of a new era, so that in a few years U.S. students will outshine their global competitors.

Meanwhile, I leave you with the following exercise.

The square of 24 in base b equals 554 in base b. What is base b?

It is not required that you solve it, but if you think you’ve got the answer, send it to me and I’ll tell you if you are right. Extra credit if you show your work. But I wouldn’t give this problem to the average teen. Not only is it difficult, but he/she would find it of no practical use — not good for anything!

Resources

For more on math learning issues:

• Devlin, K., The Math Instinct: Why You’re a Mathematical Genius, Thunder’s Mouth Press, Avalon Publishing, 2005.

• Nunes, T., A. D. Schliemann, and D. W. Carraher, Street Mathematics and School Mathematics, Cambridge University Press, 1993.

For math history (with some opinion and whimsy added):

• Livio, M., The Golden Ratio: The Story of Pi, Broadway Books, Random House, 2002.

• Kaplan, R., The Nothing That Is: A Natural History of Zero, Oxford University Press, 2000.

• Beckmann, P., A History of Pi, Barnes & Noble, 1993.

• Szpiro, G., The Secret Life of Numbers: 50 Easy Pieces on How Mathematicians Work and Think, Joseph Henry Press (A National Academies Imprint), 2006.

For math puzzles:

• Gardner, M., The Second Scientific American Book of Mathematical Puzzles and Diversions, University of Chicago Press, 1987.

• Problemcorner.org (some 20,000 problems to keep you occupied)

• TheMathForum@Drexel (K-12 math problems, puzzles, and tricks)

• MathProOnline (references to math problems)

• Mathpuzzle.com

Donald Christiansen is the former editor and publisher of IEEE Spectrum and an independent publishing consultant. He can be reached at donchristiansen@ieee.org.